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\documentclass{article}
\usepackage{enumitem}
\begin{document}
\begin{enumerate}[label=``\arabic*., ref=\arabic*]
\item bal
\item bla blala bla
\item bla bla bla''
\end{enumerate}
\end{document}
ref=\arabic* is needed so that if you label an item and \ref it later, you won't get " in the reference.

There is a known clash between bibentry and hyperref packages. The latter is loaded by your custom class.
A workaround is to delete the line
\usepackage{bibentry}
from your code and add
\RequirePackage{bibentry}
\makeatletter\let\saved@bibitem\@bibitem\makeatother
just before
\documentclass[twoside,11pt]{Latex/Classes/thesis}
and
...

The cause of problem is not easy to determine, as you've noted. It arises in the function output.nonnull, which is called by output.check, which is called by inproceedings on the line that says
format.authors "author" output.check
I suggest you do the following:
Find the function output.nonnull. (I believe it starts on line 69.)
In this function, change ...

When you do BibTeX with Case 2, you get
Database file #1: abbr.bib
I was expecting an "="---line 1 of file abbr.bib
: @string{ieee
: Transactions on Automatic Control = "IEEE trans. Automat. Contr."}
I'm skipping whatever remains of this command
What does it mean? That a string abbreviation cannot contain spaces.
Dead end. You can't have ...

If you load the natbib citation management package (with the option numbers, since it looks like you use a numeric citation call-out style), you could issue the following citation command to achieve your objective:
\cite[\S12.3.2]{<foo>}
where <foo> is the key of the bib entry you wish to cite.
Here's the output of a minimal working example ...

In your hyperref set-up you have
backref=section
Which via the hyperref manual
Adds ‘backlink’ text to the end of each item in the bibliography, as a
list of section numbers. This can only work properly if there is a
blank line after each \bibitem. Supported values are section, slide,
page, none, or false. If no value is given, section is taken ...

We can create a new command that "forgets" the shorthand field for the time being, resulting in a "normal" citation
\DeclareCiteCommand{\citenoso}
{\usebibmacro{cite:init}%
\usebibmacro{prenote}}
{\clearfield{shorthand}%
\usebibmacro{citeindex}%
\usebibmacro{cite}}
{}
{\usebibmacro{cite:postnote}}
...

biblatex has the \ifcitation and \ifbibliography tests to check whether we are in a citation or the bibliography, respectively.
In your case, we might define \pra as
\newcommand*{\pra}{%
\ifcitation
{PRA}
{Phys. Rev. A}}
such that \pra expands to "PRA" in a citation and to "Phys. Rev. A" everywhere else (especially in the bibliography).
MWE
...

It is not quite clear for which part you want to deactivate abbreviate, but the key simply (re)defines a command and so switching is not difficult:
\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage[
backend=biber
,abbreviate=false
]{biblatex}
\addbibresource{biblatex-examples.bib}
\begin{document}
...

The macro \citeA is "fragile", in the LaTeX-specific sense of the word. If it occurs inside a "moving argument" (more LaTeX jargon, sorry), such as the argument of a \caption instruction, one must prefix a \protect instruction:
\caption{\label{fig:researchscope}The process of concept-driven design approach in
relation to theory and use situation, ...

There is a similar question in StackOverflow.
The problem with the \cite command in the figure caption is that it generates conflicts with the automatic creation of the List of Figures. So the first option is to put an alternative caption into brackets:
\caption[test caption]{test caption from~\cite{MyCite}}
You can also use the \protect environnement as ...

The ieeetr bibliography style is set not to typeset the contents of the number field of an entry of type @article.
As @yo' has already pointed out, you also need to fix the spelling of the first author's surname. I suggest you use {\AA}str{\"o}m. For more on how to input accented characters in bibliographies, see the posting How to write “ä” and other ...

biblatex-historian has an option for suppressing the "see note" string:
printnoterefs=false
See p. 5 of the biblatex-historian documentation.
The title option to \printbibliography lets you change the title
\printbibliography[title=\bibname]
The default heading for articles (and article-like classes) is "References" (stored in \refname), while for ...

Like Ulrike said in a comment you can use csquotes block environments. Your attempt looks like csquotes' displaycquotes environment.
The output of those environments can be customized by redefining the hooks
\mkbegdispquote{<punct>}{<cite>}
\mkenddispquote{<punct>}{<cite>}
This will however affect all displayquote environments of ...

I suggest you do the following to get all citation call-outs and all cross-references to items (including the names of the items) colored in blue:
Load the hyperref package with the options colorlinks=true and allcolors=blue. This will assure that citation call-outs are colored in blue.
Load the cleveref package with the options nameinlink, noabbrev, and ...

Since the page-related references are specific to each entry, you can't use a single \cite command -- unless you want to risk confusing your readers...
Use the instruction
(\citeNP[pp.~15--22]{atkin14}; \citeNP[p.~282]{rub14})
to create two separate citation call-outs separated by a semicolon and encased in a single pair of round parentheses. The NP in ...

\nocite worked when I tried it, this has one visible \cite and two references in the list at the end.
(Please always provide complete documents, not just fragments)
\documentclass[
12pt,
a4paper,
openright,
%twoside,
oneside,
dvipsnames,
...

Does your MWE not work? I can compile it.
I guess there is a problem in the settings for the compilers. I used following settings and it worked:
pdflatex -synctex=1 -interaction=nonstopmode %.tex
bibtex %
pdflatex -> bibtex -> pdflatex -> pdflatex
(also using TexMaker)

The basic idea behind this answer is very similar to lockstep's answer to How do I instruct \fullcite to use maxbibnames rather than maxcitenames?, we want to change maxcitenames locally only for full citations though, so we cannot modify the entire citation command, we need to find and tweak the actual macro responsible for full citations (which is ...

If you are willing to use natbib, you get cite commands like the one you are looking for:
http://merkel.zoneo.net/Latex/natbib.php
You can use e.g. \citet*{jon90} for that. If you do not like the round brackets for citing, you can customize that (and a lot more).

Here is a way to do it, using the hyperref package with the backref option. It simply prints out the backrefs using a,b,... by incrementing a counter.
\makeatletter
\newcounter{backref@bref} %Define new counter
\long\def\hyper@letters@backref#1#2#3{ %Defines new backref printer
\stepcounter{backref@bref}%
\hyperlink{#3}{\alph{backref@bref}}% Shows ...